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2 Description of the problem

 The problem to be solved is the following: Given (photometric) observations of a pair of interacting galaxies, as well as systemic velocities for the two galaxies, determine the parameters of the orbit.

In principle, three observations at different times suffice to deduce the orbital parameters of a comet or an asteroid. For galaxies, however, one can only obtain observations of a single snapshot. Fortunately, such snapshots contain a wealth of information since the gravitational forces between the galaxies produce deformations in the form of, for example, arms, tails, and bridges connecting the galaxies. In addition to the position data, the radial velocity field can also be measured. From the complete set of data, information about scale radii, scale heights, disc inclinations, velocity dispersions, and masses can be obtained.

However, positions along the line of sight and velocities in the plane of the sky cannot be measured, and the problem of finding the orbital parameters is therefore far from trivial. Additional complications appear since observations never provide perfect, noise-free data.

A snapshot from a simulation of two face-on galaxies is shown in the left panel of Fig. 1. Clearly, the galaxies are strongly interacting with material being torn off from both. Just looking at the snapshot, it is difficult to say very much about the orbit. Given systemic velocities, positional information of the type shown in Fig. 1, and, in some cases, the radial velocity field, I shall in Sects. 4 and 5 use a GA to determine the orbital parameters of several (artificial) interacting systems.

The purpose of the paper is to test GAs as a method for finding orbital parameters of interacting galaxies, rather than testing the methods used for the individual simulations. In order to study the more violent types of interactions (e.g. mergers) where even the inner regions of the galaxies are strongly deformed, fully self-gravitating simulations, in which the mass distributions of the two galaxies are taken as unknown variables, would be required.

However, in this paper, the discussion will be restricted to less violent types of interactions, for which some simplifications could be made:

First, the scale length ($r_{\rm d}$), inclination, (i) and major axis position angle (PA) of each disc were assumed to be known from measurements of the central regions of the galaxies.

Second, the simulations were non-self-gravitating, i.e. interparticle forces were neglected, the disc particles were influenced only by the gravitational forces from the two point particles (of mass m1 and m2), representing each galactic disc, and no dispersion velocities were added. This was the case both for the simulations carried out in order to determine orbital parameters, and for the simulations which generated the artificial observational data.

For the former, it would have been very difficult to use self-gravitating simulations, due to the large number of simulations which had to be carried out for each determination of orbital parameters.

For the latter, self-gravitating simulations could, in principle, have been employed, but it would not have been useful to do so, since the use of self-gravity would introduce a number of complications having, in general, little to do with the performance of the GA. For instance, in a self-gravitating simulation, velocity dispersions must be added and it may also be necessary to add a stabilizing halo, thus introducing several additional model parameters. Also, if a grid code were to be employed, the additional softening caused by the finite size of the grid cells would complicate the comparison even more.

The use of non-self-gravitating simulations for generating the artificial data does, however, somewhat simplify the task of the GA. In such cases, it is possible for the GA to arrive at orbital parameters such that the artificial data are reproduced exactly. This, of course, would neither hold for real data nor for data generated by a simulation incorporating self-gravity, where instead the results corresponding to the best orbital parameters found by the GA would only approximate the observational data, rather than reproducing them exactly.

\psfig {,height=8.5cm}\end{figure} Figure 1: Computation of the data for an observation of a pair of interacting galaxies. The grid is superposed on the image of the interacting galaxies, and the grid cells are assigned values corresponding to the mass in each cell. For clarity, only a very coarse grid has been used in this figure

The neglect of self-gravity may not be a serious limitation in the cases where only the outer parts of the two galaxies are significantly affected by the interaction: In a tenuous, low mass arm or tail consisting of material from the outer regions of either galaxy, the self-gravity is usually less important than the tidal field of the two galaxies. The discussion below will deal with such systems, and more violent forms of interactions will not be considered.

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